Abstract: 1. Random Operators in Hilbert Space.- 1. Basic Definitions.- 1.1 Strong Random Operator.- 1.2 Weak Random Operator.- 1.3 Product of Random Operators.- 2. Characteristic Functions of Random Operators.- 2.1 Definition.- 2.2 Characteristic Functions of Strong and Bounded Operators.- 2.3 Gaussian Random Operators.- 3. Convergence of Random Operators.- 3.1 Weak Convergence of Random Operators.- 3.2 Strong Convergence of Random Operators.- 3.3 Convergence of Distributions corresponding to Random Operators.- 2. Functions of Random Operators.- 4. Spectral Representation for Symmetric Random Operators.- 4.1 Symmetric Random Operators and Selfadjoint Extensions.- 4.2 Spectral Representation of a Selfadjoint Random Operator.- 4.3 Spectral Representation of a Strong Symmetric Operator.- 5. Equations with Symmetric Random Operators.- 5.1 Evolution Equations.- 5.2 Schrodinger-type Equations.- 5.3 Spectral Moment Functions.- 5.4 Equation of Fredholm Type.- 6. Equations with Semi-Bounded Random Operators.- 6.1 Nonnegative Closed Random Operators.- 6.2 Resolvent of a Nonnegative Operator.- 6.3 Resolvent of a Nonnegative Random Operator.- 6.4 Equations of Fredholm Type.- 6.5 Equations of Evolution Type.- 3. Operator-Valued Martingales.- 7. Operator-Valued Martingale Sequences.- 7.1 Weak Operator-valued Martingale.- 7.2 Strong Operator-valued Martingales.- 7.3 Operator-valued Martingale.- 8. Convergence of Infinite Products of Independent Random Operators.- 8.1 Infinite Products as Martingales.- 8.2 Convergence of Infinite Products given the Existence of Two Moments.- 8.3 Convergence of Infinite Products in Absolute Norm.- 9. Continuous Operator-Valued Martingales.- 9.1 Some Properties of Continuous Real-valued Local Martingales.- 9.2 Continuous Martingales with values in X.- 9.3 Operator-valued Continuous Martingales.- 9.4 Strong Operator-valued Wiener Processes.- 4. Stochastic Integrals and Equations.- 10. Stochastic Integrals with Respect to an X-Valued Martingale.- 10.1 Definition.- 10.2 Integrals for Processes with Regular Characteristics.- 10.3 Stochastic Integral with respect to a Wiener Process.- 11. Stochastic Integral with Respect to an Operator-Valued Martingale.- 11.1 Integrals of X-valued Functions.- 11.2 Integrals of Operator-valued Functions.- 12. Stochastic Operator Equations.- 12.1 Operator-valued Functions of Random Operators.- 12.2 Stochastic Equations Involving I(Z, Y)t.- 12.3 Stochastic Equations Involving I*(Z, Y)t.- 12.4 Some Generalizations.- 5. Linear Stochastic Operator Equations.- 13. Generalization of the Stochastic Operator Integral.- 13.1 General Form of the Linear Equation.- 13.2 A Generalization of the Stochastic Integral.- 14. Linear Differential Operator Equations.- 14.1 Definition of a Linear Equation.- 14.2 Existence of Uniqueness of Solution.- 14.3 Linear Transformations of Solutions.- 14.4 Equations for Moments of the Solution of a Stochastic Equation.- 15. Continuous Stochastic Semigroups.- 15.1 Solutions of Simple Linear Equations -Stochastic Semigroups.- 15.2 Time Reversal in Stochastic Differential Equations.- 15.3 Definition of Stochastic Semigroups.- 15.4 Semigroups which are Martingales.

Abstract: Cartesian Gaussian functions are employed to derive general expressions for integrals over all one‐electron operators of the Breit–Pauli Hamiltonian. It is shown that in atoms of higher atomic number p6, p8, ⋅⋅⋅ operators can be important in determining relativistic corrections to the kinetic energy. All other operators of this Hamiltonian can be expressed as some derivative of 1/r. Thus, a general expression is derived for the integral over the operator (∂1/∂x1) (∂m/∂ym) (∂n/∂zn) (1/r) by employing its Fourier transform. The operator and charge‐distribution‐dependent parts can be separately identified in the resulting expression and hence for a given charge distribution, integrals over any number of operators that can be expressed in the above form can be obtained simultaneously. In addition to nuclear attraction, these operators include the spin‐orbit and Darwin terms of the Breit‐Pauli Hamiltonian, as well as the electric field components and their derivatives, and other interactions over operators req...

Abstract: In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space H2( S) associated with the Shilov boundary S of a bounded symmetric domain D in C' of arbitrary rank. The Jordan triple system Z > C'" associated with D is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent e E Z induces an irreducible representation ("e-symbol") of the C*- algebra ,' generated by all Toeplitz operators Tf with continuous symbol function f. 0. Introduction. Toeplitz operators on the boundary T = aA of the open unit disc A C C play an important role in function theory of one complex variable (cf. (9, Chapter 7)). In several dimensions Toeplitz operators have been mainly studied for strictly pseudo-convex domains D C C " (4, 13, 14, 24, 28), in particular for the Hilbert ball (7), using the relationship with pseudo-differential operators. Another class of domains in C" generalizing the unit disc is the class of bounded symmetric domains (Cartan domains and exceptional domains), which have a more complicated boundary structure compared to the strictly pseudo-convex case. The aim of this paper is the study of Toeplitz operators T1 with continuous symbol functionf E- C(S) on the Shilov boundary S of a bounded symmetric domain D of arbitrary rank r. In the special cases of the Hilbert ball (r= 1) and the Lie ball (r = 2), the structure of the operators TI and of the Toeplitz C*-algebra

Abstract: Algebras of right invariant pseudo-differential operators are constructed on any graded nilpotent group G . We obtain parametrices in such algebras for right invariant differential operators P such that P and its adjoint satisfy the hypoellipticity condition of Rockland .

Abstract: A Jordan canonical form for formal difference operators, like the one in [7], is derived in a way inspired by [3], [4]. This yields a classification of meromorphic difference operators in a neighbourhood of infinity, up to formal equivalence.

Abstract: We construct a complete system of unitary invariants for some classes of commuting operators which include finite matrices, integral and differential operators. These invariants closely connected with the theory of determinantal curves and related vector bundles.

Abstract: A perturbation theory for determinants of differential operators regularized through the ζ-function technique is presented. The application of this approach to the study of chiral changes in the fermionic path-integral variables is discussed.

Abstract: Some properties of the one-to-one mapping AA( I A*A)I , 2of the pure contractions onto the closed and densely-defined operators are proved, in particular that it commutes with adjunction and preserves normalitv. In a Hilbert space H, a pure contraction is a linear operator A on H such that 1 Ax 1< I Ix II for all x # 0 in H. In [2] I showed that the function F defined by F(A) = A(1 A*A)-I/2 maps the set of all pure contractions one-to-one onto the set of all closed and densely-defined operators in H. Since F preserves many properties of operators, it may be used to reformulate questions about unbounded operators in terms of bounded ones. For example, in [1, ?5, pp. 708-712], Cordes and Labrousse prove that if a closed and densely-defined operator C is semiFredholm then so is the bounded operator C(1 + C*C)-f/2; the latter is simply F `(C). The purpose of the present report is to indicate some further properties of F, and to establish some connections between unbounded symmetric operators and certain hyponormal contractions, leading to a reduction of the problem of finding selfadjoint extensions of symmetric operators to a corresponding problem involving only bounded operators. From this point on, A denotes a pure contraction, B and B* the associated defect operators (1 A*A)l/2 and (1 -AA*)'/2, respectively, and C the closed and densely-defined operator F(A) = AB-'. (Note that F(A*) = A*B*-.) We take for granted the following relations proved in [2]: ran B = dom C, C* = BI-A*, B = (1 + C*C)-1/2, and (thus) C*C = B-2 1. Note also that since A is a pure contraction, B and B* are one-to-one. THEOREM 1. The operator C has an everywhere-defined and bounded inverse if and only if the operator A is invertible. PROOF. Since B is one-to-one and ran A = ran C, A is invertible if and only if C is one-to-one with range H. By the closed graph theorem, this is equivalent to C having a bounded inverse with domain H. LEMMA 1. F commutes with adjunction, i.e., F(A*) = F(A)*. Received by the editors September 16, 1981 and, in revised form, February 17, 1982. 1980 Mathematics Subject Classification. Primary 47A45, 47A65. 'This work was supported in part by an Ohio University Research Award. ? 1983 American Mathematical Society 0002-9939/82/0000-0447/$02.00

Abstract: This paper presents new developments in abstract spectral theory suitable for treating classical differential and translation operators. The methods are specifically geared to conditional convergence such as arises in Fourier expansions and in Fourier inversion in general. The underlying notions are spectral family of projections and well-bounded operator, due to D. R. Smart and J. R. Ringrose. The theory of well-bounded operators is considerably expanded by the introduction of a class of operators with a suitable polar decomposition. These operators, called polar operators, have a canonical polar decomposition, are free from restrictions on their spectra (in contrast to well-bounded operators), and lend themselves to semigroup considerations. In particular, a generalization to arbitrary Banach spaces of Stone's theorem for unitary groups is obtained. The functional calculus for well-bounded operators with spectra in a nonclosed arc is used to study closed, densely defined operators with a well-bounded resolvent. Such an operator L is represented as an integral with respect to the spectral family of its resolvent, and a sufficient condition is given for (-L) to generate a strongly continuous semigroup. This approach is applied to a large class of ordinary differential operators. It is shown that this class contains significant subclasses of operators which have a polar resolvent or generate strongly continuous semigroups. Some of the latter consist of polar operators up to perturbation by a semigroup continuous in the uniform operator topology.

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Abstract: In this paper, the authors obtain two kinds of endpoint estimates for strongly singular Calder´on-Zygmund operators. Moreover, the pointwise estimate for the sharp maximal function of commutators generated by strongly singular Calder´on- Zygmund operators and BMO functions is also established.

Abstract: For the two‐body system with short range potentials the wave operators are shown to be strongly approximatable by exponentials of bounded or even finite‐rank operators, which allow the calculation by expansion in a power series or even by some finite‐dimensional matrix algebra. It is based on a bounded strong approximation of the kinetic energy Hamiltonian. For the Coulomb potential the same kind of result is obtained for the Dollard modified wave operators. A generalization for the N‐body system is sketched.

Abstract: We discuss recent work on almost periodic differential and difference operators, especially the one-dimensional Schrodinger equation and its finite-difference analogue.

Abstract: CONTENTS Introduction Chapter I. Calculus of order-bounded operators § 1.1 Ideal spaces § 1.2 Order-bounded operators Chapter II. A criterion for integral representability of linear operators § 2.1 Integral operators § 2.2 Proof of the criterion for integral representability § 2.3 Some applications of the criterion for integral representability § 2.4 Calculus of order-bounded operators and the Schrodinger operator Chapter III. Compact operators in spaces of measurable functions § 3.1 The problem of majorization for compact operators § 3.2 Proof of the majorization theorem for compact operators Appendix. The absence of the property of order-boundedness for singular integral operators References

Abstract: It is shown that the Dirac equation admits a natural algebra P of global pseudodifferential operators, characterized] by the property that the “Heisenberg representation” A→exp(iHt)A exp(-iHt)=At leaves P invariant. (For a general 4×4-matrix A of pseudodifferential operators one expects A to be a matrix of Fourier integral operators). An attempt to work with P as an algebra of observables would require certain modifications of most standard dynamical observables. An Egorov-type theorem, (theorem 3.1) can be applied to gain direct insights into particle orbits spin propagation not available with other algebras. Some inconsistencies of classical theory, like “Zitterbewegung”, non-commutativity of velocity components, Klein's paradox, would naturally disappear.

Abstract: The use of closed form operators and the corresponding power series expanded operators in Foldy-Wouthuysen transformations is discussed. It is shown that in general different transformed hamiltonians arise. This is demonstrated using the free electron part of the Dirac hamiltonian; the operators and their expectation values using hydrogenic functions are considered.